I Need Help On A Couple Of Q's! (1 Viewer)

[Nerd Queen]

Hey guys,

I'm sorry to be a real pain but there is a couple of questions i am having trouble with. If anyone can help me it would be greatly appreciated!

okay so here they are (they are from maths in focus by the way)

Q1. (Q12 of 7.2)
A population in a certain city is growing at a rate proportional to the population itself. After 3 years the population increases by 20%. How long will it take for the population to double? (answer is11.4 years)

I DUNNO WHATS GOING ON HERE! I'M JUST DRAWING A BLANK!


and


Q2. (Q199(b) of test yourself 2)
okay, so it says: A 5m length of timber is used to border a triangular garden bed, with the other sides of the garden against the house walls.

(a)show that the area of the garden is 1/2x./(25-x^2)

x = these are unknowns by the way not times

anyway i did this easy enough but the next bit....

(b) Find the greatest possible area of the garden bed

i dont know what i am doing wrong i keep getting 3.5355(4dp), but the answer says 6.25m^2. I really dont know where i went wrong.


if anyone could help me on either of these questions i would be very grateful.

 

[acmilan]

Hey guys,

Q1. (Q12 of 7.2)
A population in a certain city is growing at a rate proportional to the population itself. After 3 years the population increases by 20%. How long will it take for the population to double? (answer is11.4 years)

I DUNNO WHATS GOING ON HERE! I'M JUST DRAWING A BLANK!

Ok so they say it grows at a rate proportional to the population, hence you can derive an equation for the population (lets say 'P') in terms of its intial population (A) and growth rate at a time t. The description they give mathematically translates to dP/dt = kP (where k is a constant which is unknown at this time). So you need to integrate that to get a formula for P. You 'flip' it so that it is in a form to integrate, ie. dt/dP = 1/(kP). Now if you integrate that, you get t = (1/k)*ln(P) + C (where C is a constant). Now you want to make P the subject, done so as follows:

ln(P) = kt - kC
P = e^kt.e^-kC
Let A = e^-kC (since k and C are constants, A will be constant)

Hence P = Ae^kt

Given this formula do you know how to go from there? If not please say and I or another member will specify further.

 

[acmilan]

Q2. (Q199(b) of test yourself 2)
okay, so it says: A 5m length of timber is used to border a triangular garden bed, with the other sides of the garden against the house walls.

(a)show that the area of the garden is 1/2x./(25-x^2)

x = these are unknowns by the way not times

anyway i did this easy enough but the next bit....

(b) Find the greatest possible area of the garden bed

i dont know what i am doing wrong i keep getting 3.5355(4dp), but the answer says 6.25m^2. I really dont know where i went wrong.

1/2x./(25-x^2)

does that mean 0.5x/(25-x²) ie 0.5x divided by (25-x²)?

 

[Nerd Queen]

ln(P) = kt - kC
P = e^kt.e^-kC
Let A = e^-kC (since k and C are constants, A will be constant)

Hence P = Ae^kt

Given this formula do you know how to go from there? If not please say and I or another member will specify further.

Yeah i get the formula, i even tried to use it b4 but what is getting me is that there is no initial population, there is no A! this is what i could not understand in the first place!

sorry if that sounded 'agro' but its a stupid question and should be killed!

i don't know where to go from there

do i just make up my own population? or what?

 

[Nerd Queen]

1/2x./(25-x^2)

does that mean 0.5x/(25-x²) ie 0.5x divided by (25-x²)?

No. that means 0.5x times the squate root of (25-x^2). sorry i'm not good at typing these things out ./ - meant square root

 

[Slidey]

A population in a certain city is growing at a rate proportional to the population itself. After 3 years the population increases by 20%. How long will it take for the population to double? (answer is11.4 years)

y=Ae^(kt) (since dy/dt=Ny, N constant)
y(3)=Ae^3k=6A/5 (20% of A is A/5)
e^3k=6/5
k=1/3*ln(6/5)=.0608

When the population is double, Ae^(0.0608t)=2A, so e^(0.0608t)=2
0.0608t=ln2
t=11.4 roughly

I hope.

Regardless of whether this is correct, you should note the step:
"When the population is double, Ae^(0.0608t)=2A" Essentially, this means you don't need to know the initial population.

EDIT: Forgot to take natural log of 2

 

[Nerd Queen]

hey i just redid the population growth one and i did make up my own Initial population i can't beleive i didnt think of it b4 i only said that in my last post to be a smartass!

anyway i got the right answer now, i let A=500 and went on from there. is this how it is supposed to be done? is there any sort of special trick?

so here is what i did incase it made no sense:

P=Ae^kt
Let A= 500
P=500e^kt

In 3 years it increased 20%

120% of 500 is 600

600=500e^k(3)

etc etc etc

K=0.0060077

THERE4 eqn: P=500e^0.00608t

sub 1000 in for P (double of 500)

and solve for t

i got 11.4.

is this how it should be done?

 

[Slidey]

As I said, the initial population is inconsequential, since it will cancel itself out. It is probably better not to make it up, though.

 

[acmilan]

As for the area question, you get that the maximum occurs when x = 3.5355, this is correct but is not the answer for the area. The area is the value you get when you plug this x value into the equation A = 0.5x*sqrt(25 - x²). You will then get your answer of 6.25

 

[Slidey]

Q2. (Q199(b) of test yourself 2)
okay, so it says: A 5m length of timber is used to border a triangular garden bed, with the other sides of the garden against the house walls.

Uh. What type of triangle? Rightangle? Did you type the question properly?

(a)show that the area of the garden is xsqrt(25-x^2)/2

Suppose it is, then and we'll use that for part b:

(b) Find the greatest possible area of the garden bed

y=xsqrt(25-x^2)/2
dy/dx=(sqrt(25-x^2)-x^2/sqrt(25-x^2))/2=0 (for max)
25-x^2-x^2=0
2x^2=25
x=sqrt(25/2)
Sub into y:
sqrt(25/2).sqrt(25/2)/2=25/4=6.25

 

[Nerd Queen]

As I said, the initial population is inconsequential, since it will cancel itself out. It is probably better not to make it up, though.

I'm sorry, i dont get the way you said it, it is so confusing!

if this came up in an exam would i have to do it yourway? would i loose marks if i made it up?

do u think u could "dumb-down" your method 4 me? i mean really DUMB!

 

[Nerd Queen]

As for the area question, you get that the maximum occurs when x = 3.5355, this is correct but is not the answer for the area. The area is the value you get when you plug this x value into the equation A = 0.5x*sqrt(25 - x²). You will then get your answer of 6.25

OHHHHH! DUH!

i totally forgot to put it back in the equation! thanx i owe you BIG TIME!

 

[acmilan]

You probably shouldnt do it that way since you are assuming that the initial population is something that it might not be, even though it gives the correct answer.The formal method is assuming that the initial population is A, and doing it the way Slide_Rule did it:

P = Ae^kt

When t = 3, P(3) = Ae^3k = 1.2P(0) = 1.2A

So Ae^3k = 1.2A

e^3k = 1.2
3k = ln 1.2
k = 0.0608

Now when P = 2A (double initial population)

2A = Ae^0.0608t
e^0.0608t = 2
0.0608t = ln2
t = ln2/0.0608
t = 11.4

 

[Slidey]

It's basically exactly what you did, Nerd Queen, but instead of assuming the pop is 500, asume it is A

 

[Nerd Queen]

You probably shouldnt do it that way since you are assuming that the initial population is something that it might not be, even though it gives the correct answer.

I know I was ASSUMING but no matter what i sub'ed the answer would have been the same because it is PROPORTIONAL to it. so it wouldn't have mattered whether they werent THE actual population, because the equation is a proportional one. but i do get what you are saying it makes sense and thanx for that working out i TOTALLY get it now!

thanx again, means a lot!

 

[Nerd Queen]

It's basically exactly what you did, Nerd Queen, but instead of assuming the pop is 500, asume it is A

Yea i get it now, it is sort of what i did but i used actual numbers, i didnt think of it in those terms, the way you did. i dunno call me crazy but things are a lot easier to understand if you have actual numbers rather than make beleive ones!

but THANX heaps u helped clear things up a lot